There »re two besic ways of fashioning a piece of material into a given shape. Either you start with a solid lump and take pieces off until the required shape is obtained or you have tha material in sheet form and bend it to the required shape
It should be obvious that if the latter method is used, the sheet material must first be shaped so that after it is bent you have the correct sue and shape. If. then a component is to be made of sheet material, the des»gner must not only visualize and draw the final three-dimensional component be must also calculate and draw the shape of the component In the form that it will take when marked out on the two-dimensional sheet material.
The process of unfolding the three-dimensional 'solid* is called development.
The shapes of most engineering components are whole, or parts of. pnsms. pyramids, cylinders or cones and so this chapter deals with the development of the shapes.
PRISMS
Fig. 14/1 shows how a square prism is unfolded and its development obtained.
Notice that where there are comers in the undeveloped solid, these are shown as dotted lines in the development.
To develop a square prism with an oblique top This development, shown in Fig. 14/2 should bo self-explanatory
To develop a hexagonal prism with oblique ends (Fig. 14/3)
The height of each comer of the development is found by projecting directly from the orthographic view.
The shapes of the top and the bottom are found by projecting the true shapes of the oblique faces. The top has been found by conventional means. The true shape is projected from the elevation and transferred to the development.
The true shape of the bottom of the prism has been drawn directly on the development without projecting the true shape from the elevation. The comer between lines 2 and 3 has been produced until it meets the projectors from corners 1 and 4. The produced line ia then turned through 90° and the width. 2A. marked on.
The development of intersecting square and hexe-gonal prisma meeting at right angles (Fig. 14/4) First an orthographic drawing is made and the line of interpénétration is plotted. The development of the hexagonal prism is protected directly from the F.E. and the development of the square prism is projected directly from the plan.
Projecting from the orthographic views provides much of the information required to develop the prisms, any other information can be found on one of the orthographic views and transferred to the developments. In this case, dimensions A. à. C and d have not been projected but have been transferred with dividers
The development of Intersecting hexagonal and octagonal prisms meeting at an angle (F»g. 14/5) The method of developing these prisms is identical to that used in the previous example. This example is more complicated but the developments are still projected from one of the orthographic views, and any information which is not projected across can be found on the orthographic views and transferred to the development In this case, dimensions A. b. c D. etc have not been projected but have been transferred with dividers
If you punted the curved surface of a cylinder and, whilst the paint was wet placed the cylinder on e ftat surface and then rolled it once, the pettern that the peint left on the flat surface would be the development of the curved surface of the cylinder. Fig. 14/6 shows the shape that would evolve if the cylinder was cut obliquely at one end. The length of the development would be FID. the circumference.
The oblique face has been divided into twelve equal pans and numbered. You can see where each number will touch the flat surfaco as the cylinder is rolled.
3R0 ANGLE PROJECTION
Fig. 14/7 shows how the above idea is interpreted into an accurate development of a cylinder.
To develop a cylinder with an oblique top (Fig 14/7)
A plan and elevation of the cylinder is drawn. The plan is drvided into 12 equal sectors which are numbered. These numbers are also marked on the elevation
The circumference of the cylinder is calculated and is marked out alongside the elevation. This circumference no is divided into 12 equal parts and these parts are numbered 1 to 12 to correspond with the twelve equal sectors.
The height of the cylinder at sector 1 is projected across to the development and a line is drawn up from point 1 on the development to meet the projector.
The height of the cylinder et sectors 2 and 12 is projected across to the development and lines are drawn up from points 2 and 12 on the development to meet the projector
This process is repeated for all 12 points and the intersections are joined with a neat curve.
3RD ANGLE PROJECTION
To develop e cylinder which, in elevation, haa a circular piece cut-out (Fig. 14/9) The general method ol developing a cylinder of this nature it similar to those ahown above. The plan of the cylinder is divided into twelve equal sectors and the location of the sectors which are within the circular cutout are projected down to the F.E and across to the development.
There are some more points which must also be plotted These are 3'. 5'. 9' and 11". Their positions can be seen most easily on the F.E. and they are projected up to the plan The plan shows how far they ere away from points 3. 5. 9 and 11 and these distances, a and 6. can be transferred to the development. The exact positions of these points can then be projected across from the F.E. to the development _
3RD ANGLE PROJECTION
To develop an intersecting cylinder (Fig. 14/10) The shape of the development is determined by the shape of the line of intersection. Once this has been found, the development is found using the same methods as in previous examples.
DEVELOPMENT OFLARCER CYLINDER
3RD ANGLE PROJECTION
Fig. 14/11
DEVELOPMENT OF SMALLER CYLINDER
To develop both Interacting cylinder« (Fig. 14/11) The development» oro found using methods discussed above. Particular points to notice are:
The development of the smaller cylinder is at right angles to the £ of that cylinder;
Extra points are added to the circumference of the larger cylinder (2". 3M1* and 12') so that the development can be drawn more accurately;
The diameters of the two cylinders are different: therefore the lengths of the development are different:
Both cylinders are divided into twelve equal sectors and the points where these sectors meet the line of Intersection are different on each development.
DEVELOPMENT OFLARCER CYLINDER
3RD ANGLE PROJECTION
DEVELOPMENT OF SMALLER CYLINDER
Fig. 14/12 shows how the development of a pyramid is found. If a pyramid is tipped over so thet it lies on one of its sides end is then rolled so that each of its sides touches in turn, the development is traced out. The development is formed within a circle whose radius is equal to the true length of one of the comers of the pyramid.
Fig. 14/11
Fig 14/12 R»TRUE LENGTH OF A CORNER OF THE PYRAMID
To develop the tides of the frustum of • square pyramid (Fig. 14/13)
The true length of a comar of the pyramid can be seen in the F.E. An ere is drawn. radius equal to this true length, centre the apex of the pyramid. A second arc is drawn, radius equal to the distance from the epex of the cone to the beginning of the frustum, centre the apex of the cone. The width of one side of the pyramid, meesured at the base, is measured on the plan and this is stepped round the larger arc four times
3RD ANGLE PROJECTION
3RD ANGLE PROJECTION
To develop the sides of a hexagonal frustum if the top has been cut obliquely (Fig 14/14) The F.E. does not show the true length of a corner of the pyramid. Therefore, the true length. OL. is constructed and an arc. radius OL and centre 0. is drawn. The width of one side of the pyramid, measured at the base, is stepped eround the arc six times and the six sides of the pyramid are marked on the development
Tho F.E. does not show the true length of a corner of the pyramid; equally it does not show the true distance from 0 to any of the corners 1 to 6 However, if each of these corners is projected horizontally to the line OL (the true length of a corner), these true distances will be seen. With compass centre at 0. these distances are swung round to then appropriate corners.
1ST ANGLE PROJECTION
1ST ANGLE PROJECTION
Fig. 14/17 show« how. if a coot is tippad over and than rolled it will trace out its development. The development forms e sector of a circle whose radius is equal to the slant height of the cone. The length of the arc of the sector is equal to the circumference of the base of the cone.
If the base of the con* is divided into twelve equal sectors which are numbered from 1 to 12. the points where the numbers touch the llat surface as the cone is rolled can be seen.
To develop the frustum of a cone The plan and elevation of the cone are shown in Fig. 14/18. The plan is divided into 12 equal sectors. The arc shown as dimension A is 1 /12 of the circumference of the base of the cone.
With centre at the apex of the cone draw two arcs, one with a radius equal to the disunce from the apex to the top of the frustum (measured along the side of the cone) and the other equal to the slant height of the cone.
With dividers measure distance A and step this dimension around the larger arc 12 times. (This will not give an exact measurement of the circumference at the base of the cone but it is a good approximation.)
1ST ANGLE PROJECTION
1ST ANGLE PROJECTION
To develop the frustum of a cone that has been cut obliquely (Fig. 14/19)
Divide the plan into twelve equal sectors and number them from 1 to 12. Project these down to the F.E. and draw lines from each number to the apex A You can see where each of these lines crosses the oblique top of the frustum. Now draw the basic development of the cone and number each sector from 1 to 12 and draw a line between each number and the apex A.
The lines A1 and A7 on the F.E. are the true lanQth of the slant height of the cone. In fact all of the lines from A to each number are equal in length but. on the F.E., lines A2 to A6 end A8 to A12 are shorter than A1 and A7 because they are sloping 'inwards' towards A. The true lengths from A to the oblique top of the frustum on these lines is found by projecting horizontally across to the line A1. Here, the true length can be swung round with compasses to its respective sector and the resulting series of points joined together with a neat curve.
3RD ANGLE PROJECTION
3RD ANGLE PROJECTION
To develop a cone that has a cylindrical hole cut right through (Fig. 14/20)
This development, with one eddition. is similar to the last example. Divide the plan into twelve sectors, number them and project them up to the F.E. Draw the basic development and mark and number the sectors on this development. The points where the lines joining the apex to numbers 3, 4. 5, 9, 10 and 11 cross the hole are projected horizontally to the side of the cone. They are then swung round lo meet their respective sectors on the development.
There are four more points that need to be plotted. These are found by drawing tangents to the hole from the apex to meet the base in 6'8' and 2*12". Project these points down to the plan so that their distances from the nearest sector lino can be measured with dividers and transferred to tho development The point of tangency is then projected onto the development from the F.E. in the usual way.
1ST ANGLE PROJECTION
Fig. 14/20
PARTIAL PLAN
ONLY
Fig. 14/20
1ST ANGLE PROJECTION
10 Fig. 10 shows the elevation and partly finished plan of a truncated regular pentagonal pyramid m first angle projection.
(a) Complete the plan view, (b) Develop the surface area of the sloping sides Cambridge Local Examinations
13. Make an accurate development of the sheet metal adaptor piece which is pad of the surface of a right circular cone as shown in Fig. 13. The seam is at the position marked GH. Cambridge Local Examinations
11. Fig. 11 shows two views of an oblique regular hexagonal pyramid Draw, full size: (a) the given views, and (b) the development of the sloping faces only, taking 'AG' as the joint line. Show the development in one piece Associated Examining Board
12. Draw the development of the curved s»de of the frustum of the cone, shown in Fig. 12. below the cutting plane RST Take J J es the joint line for the development
Associated Examining Board
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